Difference between VSE Add Blend Type and Compositor Mix Node Set to Add
Most likely the difference is not how blender does the add operation, but the fact that you are dealing with two very different kinds of images.


  "I personally prefer the compositor's look"


In the compositor you have the original rendered layers, 32bit float linear, scene referred information. 

If you compress that into a video format then you've transformed your images into something quite different: 8 bit display referred images that have been converted using the view transform (set in the color management section), and on top of that they have been compressed using some codec that basically trows away a lot of information in order to have smaller files and creates a new set of problems...

What is the solution then? 


  "I do not want to loose more quality in the original footage"


You don't want to loose quality on the VFX created in blender either! like the quantization and saturation errors you show on the last image. 

So instead of rendering to a video format, save your elements to OpenEXR files. You will preserve all of the information from your original render and it will give you even more flexibility for your final compositing.

Also, since you are dealing with emissive elements with no opacity, you need a format that can do proper associated alpha. EXR will allow you to use proper transparency for the emissions and glows.